A131300 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) with n>3, a(0)=1, a(1)=2, a(2)=3, a(3)=7.
1, 2, 3, 7, 14, 27, 49, 86, 147, 247, 410, 675, 1105, 1802, 2931, 4759, 7718, 12507, 20257, 32798, 53091, 85927, 139058, 225027, 364129, 589202, 953379, 1542631, 2496062, 4038747, 6534865, 10573670, 17108595, 27682327, 44790986, 72473379, 117264433, 189737882
Offset: 0
Examples
a(4) = 14 = (1 + 1 + 7 + 4 + 1).
Links
- Nathaniel Johnston, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).
Programs
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Magma
/* By first comment: */ [&+[3*Binomial(n-Floor((k+1)/2), Floor(k/2))-2: k in [0..n]]: n in [0..37]]; // Bruno Berselli, May 03 2012
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Maple
seq(add(3*binomial(floor((n+k)/2),k)-2,k=0..n),n=0..50); # Nathaniel Johnston, Jun 29 2011
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Mathematica
LinearRecurrence[{3, -2, -1, 1}, {1, 2, 3, 7}, 38] (* Bruno Berselli, May 03 2012 *) -
Maxima
makelist(expand(3*((1+sqrt(5))^(n+2)-(1-sqrt(5))^(n+2))/(2^(n+2)*sqrt(5))-2*(n+1)), n, 0, 37); /* Bruno Berselli, May 03 2012 */
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PARI
Vec((1-x-x^2+3*x^3)/((1-x-x^2)*(1-x)^2)+O(x^38)) \\ Bruno Berselli, May 03 2012
Formula
From Bruno Berselli, May 03 2012: (Start)
G.f.: (1-x-x^2+3*x^3)/((1-x-x^2)*(1-x)^2).
a(n) = (3*A131269(n)-n-1)/2.
a(n) = 3*((1+sqrt(5))^(n+2)-(1-sqrt(5))^(n+2))/(2^(n+2)*sqrt(5))-2*(n+1). (End)
a(n) = 3*A000045(n+2)-2*(n+1). - R. J. Mathar, Mar 24 2018
Extensions
Terms after a(9) from Nathaniel Johnston, Jun 29 2011
New definition from Bruno Berselli, May 03 2012
Comments